3.1 Design of infinite impulse response (iir) filters by frequency

The design of a digital filter is usually specified in terms of the characteristics of the signals to be passed through the filter. Inmany cases, the signals are described in terms of their frequency content. For example, even though it cannot be predicted just what aperson may say, it can be predicted that the speech will have frequency content between 300 and 4000 Hz. Therefore, a filter can bedesigned to pass speech without knowing what the speech is. This is true of many signals and of many types of noise or interference. Forthese reasons among others, specifications for filters are generally given in terms of the frequency response of the filter.

The basic IIR filter design process is similar to that for the FIR problem:

1. Choose a desired response, usually in the frequency domain;
2. Choose an allowed class of filters, in this case, the Nth-order IIR filters;
3. Establish a measure of distance between the desired response and the actual response of a member of the allowed class; and
4. Develop a method to find the best allowed filter as measured by being closest to the desired response.

This section develops several practical methods for IIR filter design. A very important set of methods is based onconverting Butterworth, Chebyshev I and II, and elliptic-function analog filter designs to digital filter designs by both the impulse-invariant method and the bilinear transformation. The characteristics of these four approximations are based oncombinations of a Taylor's series and a Chebyshev approximation in the pass and stopbands. Many results from this chapter can beused for analog filter design as well as for digital design.

Extensions of the frequency-sampling and least-squared-error design for the FIR filter are developed for the IIR filter. Severaldirect iterative numerical methods for optimal approximation are described in this chapter. Prony's method and direct numerical methodsare presented for designing IIR filters according to time-domain specifications.

The discussion of the four classical lowpass filter design methods is arranged so that each method has a sectionon properties and a section on design procedures. There are also design programs in the appendix. An experienced person can simplyuse the design programs. A less experienced designer should read the design procedure material, and a person who wants to understand thetheory in order to modify the programs, develop new programs, or better understand the given ones, should study the propertiessection and consult the references.

Rational function approximation

The mathematical problem inherent in the frequency-domain filter design problem is the approximation of a desired complexfrequency-response function $H_d\left(z\right)$ by a rational transfer function $H\left(z\right)$ with an Mth-degree numerator and an Nth-degree denominator for values of the complex variable $z$ along the unit circle of $z=e^{j\omega }$ . This approximation is achieved by minimizing an error measure between $H\left(\omega \right)$ and $H_d\left(\omega \right)$ .

For the digital filter design problem, the mathematics are complicated by the approximation being defined on the unit circle.In terms of $z$ , frequency is a polar coordinate variable. It is often much easier and clearer to formulate the problem such thatfrequency is a rectangular coordinate variable, in the way it naturally occurs for analog filters using the Laplace complexvariable $s$ . A particular change of complex variable that converts the polar coordinate variable to a rectangular coordinate variableis the bilinear transformation [link] , [link] , [link] , [link] .